The random variables \(X_1, X_2, \ldots , X_n\)  are i.i.d. which implies that the moment generating function of \(\sum _{k=1}^n a_kX_k\)  takes the form \(\prod _{k=1}^n M(a_kt)\) , where \(M(t)\)  denotes the moment generating function of the \(X_i\)  [1]}. Therefore,
\(\sum _{j=0}^{\infty } f_j(a) \frac{t^j}{j!}&=\prod _{k=1}^n M(a_kt)=\exp \Big ( K(a_1t)+K(a_2t)+\cdots +K(a_nt)\Big ),\) 
 in which \(K(t)=\log M(t)\)  denotes the cumulant generating function of the \(X_i\) . The identity \(K(t)=\sum _{\ell =1}^{\infty } \kappa _{\ell } t^{\ell }/\ell !\) , which defines the cumulant sequence \(\kappa _1, \kappa _2, \ldots \) , implies
\(\sum _{j=0}^{\infty } f_j(a) \frac{t^j}{j!}=\exp \Big (\sum _{\ell =1}^{\infty } \kappa _{\ell }\mathpzc {p}_{\ell } \frac{t^{\ell }}{\ell !} \Big )=\sum _{j=0}^{\infty } B_j( \kappa _1 \mathpzc {p}_1, \kappa _2 \mathpzc {p}_2, \ldots , \kappa _j\mathpzc {p}_j) \frac{t^j}{j!},\) 
 where \(B_j(x_1, \ldots , x_j)\)  denotes the complete Bell polynomial of degree \(j\)  in the variables \(x_1, x_2, \ldots , x_j\)  [2]} defined via the generating function
\(\sum _{j=0}^{\infty } B_j( x_1, x_2, \ldots , x_j) \frac{t^j}{j!}=\exp \Big (\sum _{\ell =1}^{\infty } x_{\ell } \frac{t^{\ell }}{\ell !} \Big ).\) 
 Comparing coefficients in the above expression and applying the identity
\(B_j(x_1, x_2, \ldots , x_j)=j !\sum _{\begin{array}{c}i_1,i_2, \ldots , i_{j}\ge 0\\ i_1+2i_2+\cdots +j i_j=j \end{array}}\prod _{s=1}^{j} \frac{x_s^{i_s}}{(s!)^{i_s} i_s!}=j!\sum _{\pi \in P(j)} \frac{x_{\pi }}{y_{\pi }}\) 
 completes the proof.
